Assume that John does not have any endowment; his only income is wage income Con
ID: 2506699 • Letter: A
Question
Assume that John does not have any endowment; his only income is wage income
Consider an economy with only John and a firm. John has a utility function U (x, l) = lnx + ln (1 - l) where x is the consumption and l is the amount of time he works. The firm can produce consumption good with production technology f(l) = l. Let w be the wage and p be the price for the consumption good. Write down John's budget constraint. Solve John's utility maximization problem and get his labor supply and demand for the consumption good. (Hint: Both depend on w/p. John solves maxx,f ln(x) + ln(l - l) subject to his budget) Solve the profit maximization of the firm and get the labor demand and supply for the consumption good. Find the equilibrium wage w* and good price p*. Compute the equilibrium labor supply l* and consumption x*. Solve the social planner's problem. And compare the labor supply and consumption with part (e). Are they the same? (Hint: social planner maxl,x U(x, l) s.t. x = f(l)).Explanation / Answer
1 a) the budget constraint eqn is,
total consumption = total income
px = wl
b)
Using lagrange multiplier, where t is lagrange constant
Maximize function L
L = lnx + ln(1-l) + t*(px-wl)
delU/delx = 1/x + pt = 0 ................(1)
delU/dell =-1/(1-l) - wt = 0 ......................(2)
px = wl ......................(3)
1/x = -pt
1/(1-l) = -wt
(1-l)/x = p/w ..............(a)
l/x = p/w ...................(b)
solve eqn a and b
x = 2w/p
l = .5
c)
for profit maximization
Q = l
profit = PQ-cost
= pl-wl
it is an increasing equation, so
no as l increases profit also increases
labour supply equation,
L = .5
Profit = p-.5w
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